It can be shown that \(\mathsf {CMSE}_t\)  is a convex function of \(\mathbf {\tilde{x}}_t\)  as the Hessian matrix of \(\mathsf {CMSE}_t\)  is \(2 \times \mathbf {1}\mathbf {1}^\top \) , which is positive semi-definite.
Since any local minimum of a convex function is also a global minimum [1]}, we only need to find a local minimum of \(\mathsf {CMSE}_t\) .
Then, by letting the derivative equal to zero, we have
\(\frac{\partial \mathsf {CMSE}_t}{\partial \mathbf {\tilde{x}}_t} = 2 \times \mathbf {1}\mathbf {1}^\top \mathbf {\tilde{x}}_t - 2 \times \mathbf {1}\mathbf {1}^\top \int _{\mathbb {R}^K} \mathbf {x}_t f\left(\mathbf {x}_t \vert \lbrace y\rbrace ^t_0\right) \mathrm {d} \mathbf {x}_t = \mathbf {0}.\) 
